2.4 Combining Second Quantization and Slater–Condon Rules
27
For the state (II), the operator pair c
†
a c k can be commuted to the position k without
involving a sign change, as is indicated on the right-hand side of the second line.
To proceed, we commute the two operators c
†
j and c
†
l in | 0 to the left. This goes
with an unspecified number of sign changes, indicated by the phase factor (−1)
ν ,
depending on the relative positions of j and l:
(I ) = c j c
†
j c
†
l
k
-
-
(−1)
ν
(I I ) = c l c
†
j c
†
l
a
-
-
(−1)
ν
Since the same phase factor arises in both (I) and (II), it will drop out in forming the
matrix element. Straightforward operator algebra then yields
(I ) = c
†
l
k
-
-
(−1)
ν
(I I ) = c
†
j
a
-
-
(−1)
ν
(−1)
where the sign (−1) in the second line arises from (anti-) commuting the operators
c l and c
†
j . Now we have reached a form where the two product states differ exactly
at two positions. Taking the resulting sign into account, SC rule c3 gives
N −1
j
| ˆ
V |
N −1
akl = −V lk[ ja]
(2.35)
In an analogous way, the case j = k (or j = l) can be treated. Here, the two states
can readily be reshaped such that they differ exactly at one position, and the SC rule
c2 applies:
N −1
k
| ˆ
V |
N −1
akl = −
i =k
V il[ia]
(2.36)
It is interesting to note that the corresponding matrix element of the full hamiltonian,
ˆ
H = ˆ
T + ˆ
V , is simply given by
N −1
j
| ˆ
H |
N −1
akl = −V lk[ ja]
(2.37)
comprising the case j = k(l), provided the one-particle states are HF orbitals (see
Exercise 2.2).
2.5 Change of the One-Particle Representation
The SQ creation and destruction operators are defined with respect to a given choice
of one-particle states (orbitals). This means that a change of the underlying oneparticle basis will result in a corresponding transformation of the fermion operators.
27
For the state (II), the operator pair c
†
a c k can be commuted to the position k without
involving a sign change, as is indicated on the right-hand side of the second line.
To proceed, we commute the two operators c
†
j and c
†
l in | 0 to the left. This goes
with an unspecified number of sign changes, indicated by the phase factor (−1)
ν ,
depending on the relative positions of j and l:
(I ) = c j c
†
j c
†
l
k
-
-
(−1)
ν
(I I ) = c l c
†
j c
†
l
a
-
-
(−1)
ν
Since the same phase factor arises in both (I) and (II), it will drop out in forming the
matrix element. Straightforward operator algebra then yields
(I ) = c
†
l
k
-
-
(−1)
ν
(I I ) = c
†
j
a
-
-
(−1)
ν
(−1)
where the sign (−1) in the second line arises from (anti-) commuting the operators
c l and c
†
j . Now we have reached a form where the two product states differ exactly
at two positions. Taking the resulting sign into account, SC rule c3 gives
N −1
j
| ˆ
V |
N −1
akl = −V lk[ ja]
(2.35)
In an analogous way, the case j = k (or j = l) can be treated. Here, the two states
can readily be reshaped such that they differ exactly at one position, and the SC rule
c2 applies:
N −1
k
| ˆ
V |
N −1
akl = −
i =k
V il[ia]
(2.36)
It is interesting to note that the corresponding matrix element of the full hamiltonian,
ˆ
H = ˆ
T + ˆ
V , is simply given by
N −1
j
| ˆ
H |
N −1
akl = −V lk[ ja]
(2.37)
comprising the case j = k(l), provided the one-particle states are HF orbitals (see
Exercise 2.2).
2.5 Change of the One-Particle Representation
The SQ creation and destruction operators are defined with respect to a given choice
of one-particle states (orbitals). This means that a change of the underlying oneparticle basis will result in a corresponding transformation of the fermion operators.